Sistemas y Señales Biomédicos

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2025-03-06

Sistemas y Señales Biomedicos - SYSB

Frequency Content

Introduction

  • Signals can be analyzed in both time domain and frequency domain.
  • The frequency content of a signal describes how different frequency components contribute to the overall signal.
  • Applications in biomedical signals, audio processing, communications, and image processing.

Convolution in Time Domain

  • Convolution is a fundamental operation in signal processing.

  • Given two signals ( x(t) ) and ( h(t) ), their convolution is defined as:

    \[ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau \]

  • In discrete-time, convolution is:

    \[ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \]

Convolution Theorem

  • Convolution in time domain corresponds to multiplication in frequency domain:

    \[ X(f) H(f) = Y(f) \]

  • This property is crucial in filter design and system analysis.

Introduction to Fourier Series

(-1.0, 4.0)
(-1.0, 4.0)

Introduction to Fourier Series

  • Fourier series represents periodic signals as a sum of sinusoids:

    \[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t} \]

    where ( C_n ) are the Fourier coefficients.

  • Decomposing a signal into sinusoidal components allows frequency analysis.

Fourier Coefficients

  • The Fourier coefficients ( C_n ) are computed as:

    \[ C_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt \]

  • Determines how much of each frequency is present in the signal.

Example of Fourier Series Expansion

Relationship Between Frequency Content and Sampling Frequency

  • Sampling theorem: A signal must be sampled at a frequency at least twice its highest frequency component:

    \[ f_s \geq 2 f_{max} \]

  • Aliasing occurs if sampling frequency is too low.